Abstract
In this paper, we use the notion of a mixed weakly monotone pair of maps of Gordji et al. (Fixed Point Theory Appl. 2012:95, 2012) to state a coupled common fixed point theorem for maps on partially ordered Smetric spaces. This result generalizes the main results of Gordji et al. (Fixed Point Theory Appl. 2012:95, 2012), Bhaskar, Lakshmikantham (Nonlinear Anal. 65(7):13791393, 2006), Kadelburg et al. (Comput. Math. Appl. 59:31483159, 2010) into the structure of Smetric spaces.
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1 Introduction and preliminaries
There are many generalized metric spaces such as 2metric spaces [1], Gmetric spaces [2], {D}^{\ast}metric spaces [3], partial metric spaces [4] and cone metric spaces [5]. These notions have been investigated by many authors and various versions of fixed point theorems have been stated in [6–23] recently. In [24], Sedghi, Shobe and Aliouche have introduced the notion of an Smetric space and proved that this notion is a generalization of a Gmetric space and a {D}^{\ast}metric space. Also, they have proved some properties of Smetric spaces and some fixed point theorems for a selfmap on an Smetric space. An interesting work that naturally rises is to transport certain results in metric spaces and known generalized metric spaces to Smetric spaces. In this way, some results have been obtained in [24–26].
In [27], Gordji et al. have introduced the concept of a mixed weakly monotone pair of maps and proved some coupled common fixed point theorems for a contractivetype maps with the mixed weakly monotone property in partially ordered metric spaces. These results give rise to stating coupled common fixed point theorems for maps with the mixed weakly monotone property in partially ordered Smetric spaces.
In this paper, we use the notion of a mixed weakly monotone pair of maps to state a coupled common fixed point theorem for maps on partially ordered Smetric spaces. This result generalizes the main results of [6, 27, 28] into the structure of Smetric spaces.
First we recall some notions, lemmas and examples which will be useful later.
Definition 1.1 [[24], Definition 2.1]
Let X be a nonempty set. An Smetric on X is a function S:{X}^{3}\u27f6[0,\mathrm{\infty}) that satisfies the following conditions for all x,y,z,a\in X:

1.
S(x,y,z)=0 if and only if x=y=z.

2.
S(x,y,z)\le S(x,x,a)+S(y,y,a)+S(z,z,a).
The pair (X,S) is called an Smetric space.
The following is an intuitive geometric example for Smetric spaces.
Example 1.2 [[24], Example 2.4]
Let X={\mathbb{R}}^{2} and d be an ordinary metric on X. Put
for all x,y,z\in {\mathbb{R}}^{2}, that is, S is the perimeter of the triangle given by x, y, z. Then S is an Smetric on X.
Lemma 1.3 [[24], Lemma 2.5]
Let (X,S) be an Smetric space. Then S(x,x,y)=S(y,y,x) for all x,y\in X.
The following lemma is a direct consequence of Definition 1.1 and Lemma 1.3.
Lemma 1.4 [[25], Lemma 1.6]
Let (X,S) be an Smetric space. Then
and
for all x,y,z\in X.
Definition 1.5 [[24], Definition 2.8]
Let (X,S) be an Smetric space.

1.
A sequence \{{x}_{n}\}\subset X is said to converge to x\in X if S({x}_{n},{x}_{n},x)\to 0 as n\to \mathrm{\infty}. That is, for each \epsilon >0, there exists {n}_{0}\in \mathbb{N} such that for all n\ge {n}_{0} we have S({x}_{n},{x}_{n},x)<\epsilon. We write {x}_{n}\to x for brevity.

2.
A sequence \{{x}_{n}\}\subset X is called a Cauchy sequence if S({x}_{n},{x}_{n},{x}_{m})\to 0 as n,m\to \mathrm{\infty}. That is, for each \epsilon >0, there exists {n}_{0}\in \mathbb{N} such that for all n,m\ge {n}_{0} we have S({x}_{n},{x}_{n},{x}_{m})<\epsilon.

3.
The Smetric space (X,S) is said to be complete if every Cauchy sequence is a convergent sequence.
From [[24], Examples on p.260] we have the following.Example 1.6

1.
Let ℝ be a real line. Then
S(x,y,z)=xz+yzfor all x,y,z\in \mathbb{R} is an Smetric on ℝ. This Smetric is called the usual Smetric on ℝ. Furthermore, the usual Smetric space ℝ is complete.

2.
Let Y be a nonempty subset of ℝ. Then
S(x,y,z)=xz+yzfor all x,y,z\in Y is an Smetric on Y. Furthermore, if Y is a closed subset of the usual metric space ℝ, then the Smetric space Y is complete.
Lemma 1.7 [[24], Lemma 2.12]
Let (X,S) be an Smetric space. If {x}_{n}\to x and {y}_{n}\to y, then S({x}_{n},{x}_{n},{y}_{n})\to S(x,x,y).
Definition 1.8 [24]
Let (X,S) be an Smetric space. For r>0 and x\in X, we define the open ball {B}_{S}(x,r) and the closed ball {B}_{S}[x,r] with center x and radius r as follows:
The topology induced by the Smetric or the Smetric topology is the topology generated by the base of all open balls in X.
Lemma 1.9 Let \{{x}_{n}\} be a sequence in X. Then {x}_{n}\to x in the Smetric space (X,S) if and only if {x}_{n}\to x in the Smetric topological space X.
Proof It is a direct consequence of Definition 1.5(1) and Definition 1.8. □
The following lemma shows that every metric space is an Smetric space.
Lemma 1.10 Let (X,d) be a metric space. Then we have

1.
{S}_{d}(x,y,z)=d(x,z)+d(y,z) for all x,y,z\in X is an Smetric on X.

2.
{x}_{n}\to x in (X,d) if and only if {x}_{n}\to x in (X,{S}_{d}).

3.
\{{x}_{n}\} is Cauchy in (X,d) if and only if \{{x}_{n}\} is Cauchy in (X,{S}_{d}).

4.
(X,d) is complete if and only if (X,{S}_{d}) is complete.
Proof

1.
See [[24], Example (3), p.260].

2.
{x}_{n}\to x in (X,d) if and only if d({x}_{n},x)\to 0, if and only if
{S}_{d}({x}_{n},{x}_{n},x)=2d({x}_{n},x)\to 0,
that is, {x}_{n}\to x in (X,{S}_{d}).

3.
\{{x}_{n}\} is Cauchy in (X,d) if and only if d({x}_{n},{x}_{m})\to 0 as n,m\to \mathrm{\infty}, if and only if
{S}_{d}({x}_{n},{x}_{n},{x}_{m})=2d({x}_{n},{x}_{m})\to 0
as n,m\to \mathrm{\infty}, that is, \{{x}_{n}\} is Cauchy in (X,{S}_{d}).

4.
It is a direct consequence of (2) and (3).
□
The following example proves that the inverse implication of Lemma 1.10 does not hold.
Example 1.11 Let X=\mathbb{R} and S(x,y,z)=y+z2x+yz for all x,y,z\in X. By [[24], Example (1), p.260], (X,S) is an Smetric space. We will prove that there does not exist any metric d such that S(x,y,z)=d(x,z)+d(y,z) for all x,y,z\in X. Indeed, suppose to the contrary that there exists a metric d with S(x,y,z)=d(x,z)+d(y,z) for all x,y,z\in X. Then d(x,z)=\frac{1}{2}S(x,x,z)=xz and d(x,y)=S(x,y,y)=2xy for all x,y,z\in X. It is a contradiction.
Lemma 1.12 [[27], p.7]
Let (X,d) be a metric space. Then X\times X is a metric space with the metric {D}_{d} given by
for all x,y,u,v\in X.
Lemma 1.13 Let (X,S) be an Smetric space. Then X\times X is an Smetric space with the Smetric D given by
for all x,y,u,v,z,w\in X.
Proof For all x,y,u,v,z,w\in X, we have D((x,y),(u,v),(z,w))\in [0,\mathrm{\infty}) and
if and only if x=u=z, y=v=w, that is, (x,y)=(u,v)=(z,w); and
By the above, D is an Smetric on X\times X. □
Remark 1.14 Let (X,d) be a metric space. By using Lemma 1.13 with S={S}_{d}, we get
for all x,y,u,v\in X.
Lemma 1.15 [[17], p.4]
Let (X,\u2aaf) be a partially ordered set. Then X\times X is a partially ordered set with the partial order ⪯ defined by
Remark 1.16 Let X be a subset of ℝ with the usual order. For each ({x}_{1},{x}_{2}),({y}_{1},{y}_{2})\in X\times X, put {z}_{1}=max\{{x}_{1},{y}_{1}\} and {z}_{2}=min\{{x}_{2},{y}_{2}\}, then ({x}_{1},{x}_{2})\u2aaf({z}_{1},{z}_{2}) and ({y}_{1},{y}_{2})\u2aaf({z}_{1},{z}_{2}). Therefore, for each ({x}_{1},{x}_{2}),({y}_{1},{y}_{2})\in X\times X, there exists ({z}_{1},{z}_{2})\in X\times X that is comparable to ({x}_{1},{x}_{2}) and ({y}_{1},{y}_{2}).
Definition 1.17 [[27], Definition 1.5]
Let (X,\u2aaf) be a partially ordered set and f,g:X\times X\u27f6X be two maps. We say that a pair (f,g) has the mixed weakly monotone property on X if, for all x,y\in X, we have
and
Example 1.18 [[27], Example 1.6]
Let f,g:\mathbb{R}\times \mathbb{R}\u27f6\mathbb{R} be two functions given by
Then the pair (f,g) has the mixed weakly monotone property.
Example 1.19 [[27], Example 1.7]
Let f,g:\mathbb{R}\times \mathbb{R}\u27f6\mathbb{R} be two functions given by
Then f and g have the mixed monotone property but the pair (f,g) does not have the mixed weakly monotone property.
Remark 1.20 [[27], Remark 2.5]
Let (X,\u2aaf) be a partially ordered set; f:X\times X\u27f6X be a map with the mixed monotone property on X. Then for all n\in \mathbb{N}, the pair ({f}^{n},{f}^{n}) has the mixed weakly monotone property on X.
2 Main results
Theorem 2.1 Let (X,\u2aaf,S) be a partially ordered Smetric space; f,g:X\times X\u27f6X be two maps such that

1.
X is complete;

2.
The pair (f,g) has the mixed weakly monotone property on X; {x}_{0}\u2aaff({x}_{0},{y}_{0}),f({y}_{0},{x}_{0})\u2aaf{y}_{0} or {x}_{0}\u2aafg({x}_{0},{y}_{0}),g({y}_{0},{x}_{0})\u2aaf{y}_{0} for some {x}_{0},{y}_{0}\in X;

3.
There exist p,q,r,s\ge 0 satisfying p+q+r+2s<1 and
(2.1)
for all x,y,u,v\in X with x\u2aafu and y\u2ab0v where D is defined as in Lemma 1.13;

4.
f or g is continuous or X has the following property:

(a)
If \{{x}_{n}\} is an increasing sequence with {x}_{n}\to x, then {x}_{n}\u2aafx for all n\in \mathbb{N};

(b)
If \{{x}_{n}\} is an decreasing sequence with {x}_{n}\to x, then x\u2aaf{x}_{n} for all n\in \mathbb{N}.
Then f and g have a coupled common fixed point in X.
Proof First we note that the roles of f and g can be interchanged in the assumptions. We need only prove the case {x}_{0}\u2aaff({x}_{0},{y}_{0}) and f({y}_{0},{x}_{0})\u2aaf{y}_{0}, the case {x}_{0}\u2aafg({x}_{0},{y}_{0}) and g({y}_{0},{x}_{0})\u2aaf{y}_{0} is proved similarly by interchanging the roles of f and g.
Step 1. We construct two Cauchy sequences in X.
Put {x}_{1}=f({x}_{0},{y}_{0}), {y}_{1}=f({y}_{0},{x}_{0}). Since (f,g) has the mixed weakly monotone property, we have
and
Put {x}_{2}=g({x}_{1},{y}_{1}), {y}_{2}=g({y}_{1},{x}_{1}). Then we have
and
Continuously, for all n\in \mathbb{N}, we put
that satisfy
We will prove that \{{x}_{n}\} and \{{y}_{n}\} are two Cauchy sequences. For all n\in \mathbb{N}, it follows from (2.1) that
By using (2.2) we get
That is,
Analogously to (2.4), we have
It follows from (2.4) and (2.5) that
For all n\in \mathbb{N}, by interchanging the roles of f and g and using (2.1) again, we have
By using (2.2) we get
That is,
Analogously to (2.7), we have
It follows from (2.7) and (2.8) that
For all n\in \mathbb{N}, (2.6) and (2.9) combine to give
Now we have
and
For all n,m\in \mathbb{N} with n\le m, by using Lemma 1.4 and (2.11), (2.12), we have
Similarly, we have
and
and
Hence, for all n,m\in \mathbb{N} with n\le m, it follows that
Since 0\le \frac{p+q+s}{1(r+s)}<1, taking the limit as n,m\to \mathrm{\infty}, we get
It implies that
Therefore, \{{x}_{n}\} and \{{y}_{n}\} are two Cauchy sequences in X. Since X is complete, there exist x,y\in X such that {x}_{n}\to x and {y}_{n}\to y in X as n\to \mathrm{\infty}.
Step 2. We prove that (x,y) is a coupled common fixed point of f and g. We consider the following two cases.
Case 2.1. f is continuous. We have
and
Now using (2.1) we have
Therefore,
That is,
Since 0\le r+s<1, we get S(x,x,g(x,y))=S(y,y,g(y,x))=0. That is, g(x,y)=x and g(y,x)=y. Therefore, (x,y) is a coupled common fixed point of f and g.
Case 2.2. g is continuous. We can also prove that (x,y) is a coupled common fixed point of f and g similarly as in Case 2.1.
Case 2.3. X satisfies two assumptions (a) and (b). Then by (2.3) we get {x}_{n}\u2aafx and y\u2aaf{y}_{n} for all n\in \mathbb{N}. By using Lemma 1.4 and Lemma 1.13, we have
By interchanging the roles of f and g and using (2.1), we have
Again, by using (2.1), we have
It follows from (2.13), (2.14) and (2.15) that
By using Lemma 1.7 and taking the limit as n\to \mathrm{\infty} in (2.16), we have
It implies that
Since \frac{r+q+2s}{2}<1, we have S(x,x,f(x,y))+S(y,y,f(y,x))=0, that is, f(x,y)=x and f(y,x)=y. Similarly, one can show that g(x,y)=x and g(y,x)=y. This proves that (x,y) is a coupled common fixed point of f and g. □
From Theorem 2.1, we have following corollaries.
Corollary 2.2 [[27], Theorems 2.1 and 2.2]
Let (X,\u2aaf,d) be a partially ordered metric space; f,g:X\times X\u27f6X be two maps such that

1.
X is complete;

2.
The pair (f,g) has the mixed weakly monotone property on X; {x}_{0}\u2aaff({x}_{0},{y}_{0}), f({y}_{0},{x}_{0})\u2aaf{y}_{0} or {x}_{0}\u2aafg({x}_{0},{y}_{0}), g({y}_{0},{x}_{0})\u2aaf{y}_{0} for some {x}_{0},{y}_{0}\in X;

3.
There exist p,q,r,s\ge 0 satisfying p+q+r+2s<1 and
(2.19)
for all x,y,u,v\in X with x\u2aafu and y\u2ab0v, where {D}_{d} is defined as in Lemma 1.12;

4.
f or g is continuous or X has the following property:

(a)
If \{{x}_{n}\} is an increasing sequence with {x}_{n}\to x, then {x}_{n}\u2aafx for all n\in \mathbb{N};

(b)
If \{{x}_{n}\} is an decreasing sequence with {x}_{n}\to x, then x\u2aaf{x}_{n} for all n\in \mathbb{N}.
Then f and g have a coupled common fixed point in X.
Proof It is a direct consequence of Lemma 1.10, Remark 1.14 and Theorem 2.1. □
For similar results of the following for maps on metric spaces and cone metric spaces, the readers may refer to [[6], Theorems 2.1, 2.2, 2.4 and 2.6] and [[28], Theorem 3.1].
Corollary 2.3 Let (X,\u2aaf,S) be a partially ordered Smetric space and f:X\times X\u27f6X be a map such that

1.
X is complete;

2.
f has the mixed monotone property on X; {x}_{0}\u2aaff({x}_{0},{y}_{0}) and f({y}_{0},{x}_{0})\u2aaf{y}_{0} for some {x}_{0},{y}_{0}\in X;

3.
There exist p,q,r,s\ge 0 satisfying p+q+r+2s<1 and
(2.20)
for all x,y,u,v\in X with x\u2aafu and y\u2ab0v;

4.
f is continuous or X has the following property:

(a)
If \{{x}_{n}\} is an increasing sequence with {x}_{n}\to x, then {x}_{n}\u2aafx for all n\in \mathbb{N};

(b)
If \{{x}_{n}\} is an decreasing sequence with {x}_{n}\to x, then x\u2aaf{x}_{n} for all n\in \mathbb{N}.
Then f has a coupled fixed point in X.
Proof By choosing g=f in Theorem 2.1 and using Remark 1.20, we get the conclusion. □
Corollary 2.4 Let (X,\u2aaf,S) be a partially ordered Smetric space and f:X\times X\u27f6X be a map such that

1.
X is complete;

2.
f has the mixed monotone property on X; {x}_{0}\u2aaff({x}_{0},{y}_{0}) and f({y}_{0},{x}_{0})\u2aaf{y}_{0} for some {x}_{0},{y}_{0}\in X;

3.
There exists k\in [0,1) satisfying
S(f(x,y),f(x,y),f(u,v))\le \frac{k}{2}(S(x,x,u)+S(y,y,v))(2.21)
for all x,y,u,v\in X with x\u2aafu and y\u2ab0v;

4.
f is continuous or X has the following property:

(a)
If \{{x}_{n}\} is an increasing sequence with {x}_{n}\to x, then {x}_{n}\u2aafx for all n\in \mathbb{N};

(b)
If \{{x}_{n}\} is an decreasing sequence with {x}_{n}\to x, then x\u2aaf{x}_{n} for all n\in \mathbb{N}.
Then f has a coupled fixed point in X.
Proof By choosing g=f and p=k, q=r=s=0 in Theorem 2.1 and using Remark 1.20, we get the conclusion. □
Corollary 2.5 Assume that X is a totally ordered set in addition to the hypotheses of Theorem 2.1; in particular, Corollary 2.3, Corollary 2.4. Then f and g have a unique coupled common fixed point (x,y) and x=y.
Proof By Theorem 2.1, f and g have a coupled common fixed point (x,y). Let (z,t) be another coupled common fixed point of f and g. Without loss of generality, we may assume that (x,y)\u2aaf(z,t). Then by (2.1) and Lemma 1.3, we have
Since p+2s<1, we have S(x,x,z)+S(y,y,t)=0. Then x=z and y=t. This proves that the coupled common fixed point of f and g is unique.
Moreover, by using (2.1) and Lemma 1.3 again, we get
Since p+2s<1, we get S(x,x,y)=0, that is, x=y. □
Finally, we give an example to demonstrate the validity of the above results.
Example 2.6 Let X=\mathbb{R} with the Smetric as in Example 1.6 and the usual order ≤. Then X is a totally ordered, complete Smetric space. For all x,y\in X, put
Then the pair (f,g) has the mixed weakly monotone property and
Then the contraction (2.1) is satisfied with p=\frac{1}{6} and q=r=s=0. Note that other assumptions of Corollary 2.5 are also satisfied and (1,1) is the unique common fixed point of f and g.
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Dung, N.V. On coupled common fixed points for mixed weakly monotone maps in partially ordered Smetric spaces. Fixed Point Theory Appl 2013, 48 (2013). https://doi.org/10.1186/16871812201348
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DOI: https://doi.org/10.1186/16871812201348